New posts in hypergeometric-function

Closed form of $\int e^{i\csc^2(x)}dx=\int \cos\left(\csc^2(x)\right)dx+i\int \sin\left(\csc^2(x)\right)dx$

special-functions closed-form hypergeometric-function trigonometric-integrals fresnel-integrals

Evaluate $\int _{ }^{ }\frac{1}{\sqrt{1+x^3}}dx$

calculus integration indefinite-integrals special-functions hypergeometric-function

Closed-forms for $\int_0^\infty\frac{dx}{\sqrt[3]{55+\cosh x}}$ and $\int_0^\infty\frac{dx}{\sqrt[3]{45\big(23+4\sqrt{33}\big)+\cosh x}}$

calculus integration definite-integrals hypergeometric-function pell-type-equations

Prove $_4F_3(1/8,3/8,5/8,7/8;1/4,1/2,3/4;1/2)=\frac{\sqrt{2-\sqrt2+\sqrt{2-\sqrt2}}+\sqrt{2+\sqrt2+\sqrt{2+\sqrt2}}}{2\,\sqrt2}$

calculus closed-form conjectures hypergeometric-function radicals

Integral ${\large\int}_0^1\frac{dx}{(1+x^{\sqrt2})^{\sqrt2}}$

integration definite-integrals closed-form gamma-function hypergeometric-function

An intriguing pattern in Ramanujan's theory of elliptic functions that stops?

integration definite-integrals closed-form hypergeometric-function elliptic-functions

Some hypergeometric transformation

combinatorics power-series hypergeometric-function

Evaluate hypergeometric $_6F_5\left(\{\frac12\}_3,\{1\}_3;\{\frac32\}_5;1\right)$

integration sequences-and-series definite-integrals closed-form hypergeometric-function

Solutions in terms of the hypergeometric functions

complex-analysis ordinary-differential-equations power-series special-functions hypergeometric-function

Infinite sum including Gaussian Hypergeometric Function

sequences-and-series hypergeometric-function

Hypergeometric series for $\mathrm{Cl}_2(\pi/3)$

sequences-and-series special-functions hypergeometric-function

How to prove $_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9\big)=\large \frac{3}{5^{5/6}}\,\phi^{-1}\,$ with golden ratio $\phi$?

calculus definite-integrals hypergeometric-function conjectures golden-ratio

On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$

calculus integration closed-form hypergeometric-function conjectures

How to calclulate a derivate of a hypergeometric function w.r.t. one of its parameters?

calculus derivatives special-functions closed-form hypergeometric-function

Integral $\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx$

integration improper-integrals closed-form hypergeometric-function error-function

A curious identity involving the Appell hypergeometric series.

ordinary-differential-equations hypergeometric-function

Evaluate the integral $\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$

integration definite-integrals hypergeometric-function

Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$

calculus derivatives definite-integrals special-functions hypergeometric-function

Closed form solutions for a family of hypergeometric sums.

special-functions hypergeometric-function sequences-and-series

Looking for a closed form for a ${}_4 F_3\left(\ldots,1\right)$

sequences-and-series special-functions closed-form hypergeometric-function elliptic-integrals